package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Use BIConjugate Gradient iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` and ``A^T x`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Use BIConjugate Gradient STABilized iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Use Conjugate Gradient iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. ``A`` must represent a hermitian, positive definite matrix. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Use Conjugate Gradient Squared iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real-valued N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Use Generalized Minimal RESidual iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real or complex N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : int Provides convergence information: * 0 : successful exit * >0 : convergence to tolerance not achieved, number of iterations * <0 : illegal input or breakdown

Other parameters ---------------- x0 : array, matrix Starting guess for the solution (a vector of zeros by default). tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. restart : int, optional Number of iterations between restarts. Larger values increase iteration cost, but may be necessary for convergence. Default is 20. maxiter : int, optional Maximum number of iterations (restart cycles). Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : sparse matrix, dense matrix, LinearOperator Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used. callback : function User-supplied function to call after each iteration. It is called as `callback(args)`, where `args` are selected by `callback_type`. callback_type : 'x', 'pr_norm', 'legacy', optional Callback function argument requested:

• ``x``: current iterate (ndarray), called on every restart
• ``pr_norm``: relative (preconditioned) residual norm (float), called on every inner iteration
• ``legacy`` (default): same as ``pr_norm``, but also changes the meaning of 'maxiter' to count inner iterations instead of restart cycles. restrt : int, optional DEPRECATED - use `restart` instead.

See Also -------- LinearOperator

Notes ----- A preconditioner, P, is chosen such that P is close to A but easy to solve for. The preconditioner parameter required by this routine is ``M = P^-1``. The inverse should preferably not be calculated explicitly. Rather, use the following template to produce M::

# Construct a linear operator that computes P^-1 * x. import scipy.sparse.linalg as spla M_x = lambda x: spla.spsolve(P, x) M = spla.LinearOperator((n, n), M_x)

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import gmres >>> A = csc_matrix([3, 2, 0], [1, -1, 0], [0, 5, 1], dtype=float) >>> b = np.array(2, 4, -1, dtype=float) >>> x, exitCode = gmres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True

Make a linear system Ax=b

Parameters ---------- A : LinearOperator sparse or dense matrix (or any valid input to aslinearoperator) M :

inearOperator, Nones

preconditioner sparse or dense matrix (or any valid input to aslinearoperator) x0 : array_like, None initial guess to iterative method b : array_like right hand side

Returns ------- (A, M, x, b, postprocess) A : LinearOperator matrix of the linear system M : LinearOperator preconditioner x : rank 1 ndarray initial guess b : rank 1 ndarray right hand side postprocess : function converts the solution vector to the appropriate type and dimensions (e.g. (N,1) matrix)

Decorate a function with a threading lock and prevent reentrant calls.

Use Quasi-Minimal Residual iteration to solve ``Ax = b``.

Parameters ---------- A : sparse matrix, dense matrix, LinearOperator The real-valued N-by-N matrix of the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` and ``A^T x`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : array, matrix Right hand side of the linear system. Has shape (N,) or (N,1).

Returns ------- x : array, matrix The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters ---------------- x0 : array, matrix Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior.

.. warning::

The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M1 : sparse matrix, dense matrix, LinearOperator Left preconditioner for A. M2 : sparse matrix, dense matrix, LinearOperator Right preconditioner for A. Used together with the left preconditioner M1. The matrix M1*A*M2 should have better conditioned than A alone. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

See Also -------- LinearOperator

Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import qmr >>> A = csc_matrix([3, 2, 0], [1, -1, 0], [0, 5, 1], dtype=float) >>> b = np.array(2, 4, -1, dtype=float) >>> x, exitCode = qmr(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True

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